Sheaf Cohomology BCI Skill: Local-to-Global Consistency Skill

Sheaf Cohomology BCI Skill: Local-to-Global Consistency

Status: Production Ready Trit: -1 (MINUS - validator) Color: #2626D8 (Blue) Principle: Sheaf condition = local-to-global consistency principle Frame: Cellular sheaf on BCI signal graph with Cech cohomology

Overview

Sheaf Cohomology verifies that local BCI measurements (per channel) can be consistently glued into a global signal. Implements:

Cellular sheaf: Stalks F(v) at vertices, restriction maps on edges
Sheaf condition: Gluing axiom verification across overlapping regions
Cech cohomology: H^0 (global sections), H^1 (obstructions to gluing)
Sheaf Laplacian: Tr(L_F) measures total inconsistency
Descent/gluing: Attempt global section construction, identify obstruction class

Correct by construction: H^1 = 0 iff local sections glue uniquely to global.

Core Formulae

Cellular sheaf F on graph G = (V, E):
  F(v) = stalk (vector space) at vertex v
  rho_{v,e}: F(v) -> F(e)  restriction map

Sheaf condition:
  rho_{u,e}(s_u) = rho_{v,e}(s_v) for all edges e=(u,v)

Cech cohomology:
  C^0 = prod F(U_i)           (local sections)
  C^1 = prod F(U_i cap U_j)   (overlap data)
  delta_0: C^0 -> C^1          (coboundary)
  H^0 = ker(delta_0)           (global sections)
  H^1 = ker(delta_1)/im(delta_0)  (obstructions)

Sheaf Laplacian:
  L_F = delta^T delta
  Tr(L_F) = total inconsistency
  ker(L_F) = harmonic sections (global)

Key Results

Multi-World Sheaf Comparison:
  world-a: Tr(L_F)=3.24, non-trivial obstruction (beta-gamma disagreement)
  world-b: Tr(L_F)=0.09, GLUED (consistent signals, trivial H^1)
  world-c: Tr(L_F)=20.75, highly inconsistent (diverse channels)

BCI Integration (Layer 19)

Completes the Topology Chain: L8 -> L13 -> L14 -> L17 -> L19

L8 Persistent Homology: Betti = ranks of constant sheaf cohomology
L14 Cohomology Ring: Sheaf H^k(X,F) generalizes H^k(X,Z)
L17 de Rham: de Rham = cohomology of sheaf of smooth functions
L18 Info Geometry: Sheaf of distributions, Fisher metric varies by stalk
L16 Spectral: Sheaf Laplacian L_F generalizes graph Laplacian L

Skill Name: sheaf-cohomology-bci Type: Local-to-Global Consistency / Cellular Sheaves / Cech Cohomology Trit: -1 (MINUS) GF(3): Forms valid triads with ERGODIC + PLUS skills

Integration with GF(3) Triads

stochastic-resonance (+1) x information-geometry (0) x sheaf-cohomology-bci (-1) = 0
gay-mcp (+1) x spectral-methods (0) x sheaf-cohomology-bci (-1) = 0

Agent Skills: Sheaf Cohomology BCI Skill: Local-to-Global Consistency

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